Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

10.9. Robust Concepts 653

10.9.15.For the pseudo-norm‖v‖W defined in expression (10.9.53), establish the

identity

‖v‖W=

√

3

2(n+1)

∑n

i=1

∑n

j=1

|vi−vj|, (10.9.55)

for allv∈Rn. Thuswehaveshownthat

β̂W=Argmin

∑n

i=1

∑n

j=1

|(yi−yj)−β(xci−xcj)|. (10.9.56)

Note that the formulation ofβ̂Wgiven in expression (10.9.56) allows an easy way to
compute the Wilcoxon estimate of slope by using anL 1 (least absolute deviations)
routine. Terpstra and McKean (2005) used this identity, (10.9.55), to develop R
functions for the computation of the Wilcoxon fit.

10.9.16.Suppose the random variableehas cdfF(t). Letφ(u)=

√
12[u−(1/2)],
0 <u<1, denote the Wilcoxon score function.

(a)Show that the random variableφ[F(ei)]hasmean0andvariance1.

(b)Investigate the mean and variance ofφ[F(ei)] for any score functionφ(u)

which satisfies

∫ 1

0 φ(u)du=0and

∫ 1

0 φ

(^2) (u)du=1.
10.9.17. In the derivation of the influence function, we assumed thatxwas ran-
dom. For inference, though, we consider the case thatxis given. In this case, the
variance ofX,E(X^2 ), which is found in the influence function, is replaced by its
estimate, namely,n−^1
∑n
i=1x
2
ci. With this in mind, use the influence function of
the LS estimator ofβto derive the asymptotic distribution of the LS estimator;
see the discussion around expression (10.9.24). Show that it agrees with the exact
distribution of the LS estimator given in expression (9.6.9) under the assumption
that the errors have a normal distribution.
10.9.18. As in the last problem, use the influence function of the Wilcoxon esti-
mator ofβto derive the asymptotic distribution of the Wilcoxon estimator. For
Wilcoxon scores, show that it agrees with expression (10.7.14).
10.9.19. Use the results of the last two exercises to find the asymptotic relative
efficiency (ARE) between the Wilcoxon and LS estimators ofβ.